Make a frequency distribution for the data. Then use the...

Name: Problem 41, Chapter 2: Elementary Statistics: Picturing the World
Uploaded: 2020-12-22T02:05:55-08:00
Duration: 7 min 38 s
Channel: Sneha Ravi
Description: Make a frequency distribution for the data. Then use the table to estimate the sample mean and the sample standard deviation of the data set. The distribution of the numbers of hours that a random sample of college students study per week is shown in the pie chart. Use 32 as the midpoint for "30+ hours."

Make a frequency distribution for the data. Then use the table to estimate the sample mean and the sample standard deviation of the data set. The distribution of the numbers of hours that a random sample of college students study per week is shown in the pie chart. Use 32 as the midpoint for "30+ hours."

Make a frequency distribution for the data. Then use the table to estimate the sample mean and the sample standard deviation of the data set.
The distribution of the numbers of hours that a random sample of college students study per week is shown in the pie chart. Use 32 as the midpoint for "30+ hours."

Key Concepts

Sample Standard Deviation from Grouped Data

The sample standard deviation estimated from grouped data measures the dispersion of the data around the mean. It is computed by taking the squared differences between the class midpoints and the estimated mean, weighting these differences by the frequencies, summing them, and then applying the square root after appropriate normalization.

Class Midpoints

Class midpoints are the central values of each class interval in a frequency distribution. They act as representative values for the classes when estimating measures like the mean and standard deviation from grouped data.

Sample Mean from Grouped Data

The sample mean in the context of grouped data is estimated by multiplying each class midpoint by the frequency of that class, summing these products, and then dividing by the total number of observations. This method provides an approximation of the average value in the data set.

Frequency Distribution

A frequency distribution is a table that displays the number of observations within different intervals or categories of data. This organization allows for a clear summary of how data values are spread out across the various groups.

Grouped Data

Grouped data involves organizing raw data into groups or classes, which makes it easier to analyze and interpret large data sets. Instead of handling each individual observation, grouped data provides a summarized view through intervals.

Transcript

00:01 So in this question we are given a pie chart with a bunch of values and we're asked to calculate the mean and the standard deviation for the sample.

00:12 So to do that, let's go ahead and make our frequency distribution chart.

00:19 So let's start with, since this is a grouped data, we're going to take the midpoint of each class as our value.

00:32 Of x.

00:34 So we have x.

00:38 The midpoint of 0 to 4 hours is 2.

00:43 5 to 9 is 7.

00:46 10 to 14 is 12, 15 to 19, 17, 20 to 24 is 22, 25 to 29 is 27, and for 30 plus hours we're told to list the midpoint as 32.

01:06 So we're have our values of x now we're going to put our frequencies in so we have 5 12 24 17 16 11 and 5 so we have all of our x and our frequency values now we're going to compute x times f so we have x f and for that we are going to get 5 times 2 is 10 to all times 7 is 84.

01:48 24 times 12 is 288, 17 times 17, 289, 22 times 16, 352, 352, 21, 21, actually 27, 27 times 11, we get 297, and then 32 times 5, 5, 160.

02:27 So we get x times f over here and our formula for the mean is basically the sum of x times f over our number of values.

02:41 So we're basically going to sum everything in this column and in this column and we're going to compute our mean.

02:51 And so in this column the sum is 1480.

02:57 The number of values we have is 90.

03:02 So we take 1 480 divided by 90 which gives us the sample mean of 16 .44.

03:13 So that's our first answer.

03:15 Now for the second one we need to find the standard deviation.

03:18 So let's go ahead and expand this table out.

03:25 So basically what we have to calculate here is x minus the mean of x...